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Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler B.S., Massachusetts Institute of Technology SUBMITTED in PA

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Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler B.S., Massachusetts Institute of Technology SUBMITTED in PA J- Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler B.S., Massachusetts Institute of Technology (1963) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY August, 1965 Signature of Author . .-. .. Department of Mathematics, August 31, 1965 Certified by ....-.. Thesis Supervisqr Accepted by .......... 0................................... Chairman, Departmental Committee on Graduate Students v/I 2. Cocommutative Hopf Algebras 19 with Antipode by Moss Eisenberg Sweedler Submitted to the Department of Mathematics on August 31, 1965, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Abstract In the first chapter the preliminaries of the theory of Hopf algebras are presented. The notion and properties of the antipode are developed. An important filtration is induced in the Hopf algebra by its dual when the Hopf alge- bra is split. It is shown conilpotence and an algebraically closed field insure a Hopf algebra is split. The monoid of grouplike elements is studied. In the second chapter conditions for an algebra A -- which is a comodule for a Hopf algebra H --to be of the form A 'E B ® H (linear isomorphism) are given. The dual situation is studied. The graded Hopf algebra associated with a split Hopf algebra decomposes in the above manner. Chapter III contains the cohomology theory of a commutative algebra which is a module for a cocommutative Hopf algebra. There is extension theory and specialization to the situation the Hopf algebra is a group algebra. Chapter IV is dual to chapter III. Chapter V is devoted to coconnected cocommutative Hopf algebras, mostly in characteristic p > 0 . There, the notion of divided powers is developed and shown to charac- terize the coalgebra structure of a class of Hopf algebras. The Hopf algebras are shown to be extensions of certain sub Hopf algebras by their primitive elements. Thesis Supervisor: Bertram Kostant Title: Professor of Mathematics 3. Contents Introduction ............................................ 4 Chapter I. Preliminaries ............................ 7 Chapter II. Decompositions ......................... 36 Chapter III. Cohomology ....................----. 47 Chapter IV. Cohomology .......................... 87 Chapter V. Cocommutative Coconnected Hopf Algebras. 93 Bibliography . ......................... ......0...... Index ........ ......................... 000............ 162 Biography .... ... *0..........................-.....- 166 -I Introduction A Hopf algebra as considered herein is simultaneously an algebra and a coalgebra where the algebra structure morphisms are morphisms of the coalgebra structure, (or vice versa). This differs from the graded Hopf algebras of [2] Milnor and Moore, except in characteristic 2. The problem is to determine the structure of cocommutative Hopf algebras. Our first approach lies in a cohomology theory. We have constructed abelian cohomology groups Hi(HA) , H(CH)i , 0 < i E Z where H is a Hopf algebra, A an algebra which is a left "C.H.A." H-module, C a coalgebra which is a right "C.H.A." H-comodule. We then determine the structures of algebras (coalgebras) which are extensions of H (C) by B (H) This theory applies to the algebra structure and the co- algebra structure--separately--of coconnected cocommutative Hopf algebras. We hope to develop an extension theory where the extension is a Hopf algebra and is an extension of one Hopf algebra by another. Our cohomology theory gives the familiar group co- homology in case H =p-(G) the group algebra of the group G . If Ar is the group of regular (invertible) elements 5. of A then Hi(H,A) = Hi(G,Ar) Furthermore, if A is a finite Galois extension of the underlying field k and G is the Galois group of A/k then the isomorphism classes of extensions of H =F(G) by A form a subgroup of the Brauer Group. Kostant has shown--the results are unpublished--that a split cocommutative Hopf algebra with antipode is a smash product of a group algebra and a coconnected cocommutative Hoof algebra; and that in characteristic zero a coconnected cocommutative Hopf algebra is a universal enveloping alge- bra of the Lie algebra of primitive elements. We present proofs of these results and study coconnected cocommuta- tive Hopf algebras in characteristic- p > 0 . For a cer- tain class (including all where the restricted Lie algebra of primitive elements is finite dimensional) of coconnected cocommutative Hopf algebras we are able to determine the coalgebra structure. The coalgebra structure is described in a generalization of the Poincare-Birkhoff-Witt theorem. We now outline the generalization. In characteristic zero let (x be an ordered basis for the Lie algebra L , L C U its universal enveloping algebra, a Hopf algebra. If I= i = 0,1,2,... 6. then dx( = i=oz x2=0n The Poincare-Birkhoff-Witt theorem is equivalent to: a a <--- a i) x - x m = 0, mi O < e E Z forms a basis for U In any characteristic we say xo, xlV x2,9..... n . is a sequence of divided powers if n dx =Zx.® xn 1& n-i 1=0 We show how ordered products of divided powers--as in i)-- form a basis for the certain class of coconnected cocommu- tative Hopf algebras. For all coconnected cocommutative Hopf algebras in characteristic p > 0 , we show the Hopf algebra obtained by factoring out the ideal generated by the primitive ele- ments is isomorphic to a sub Hopf algebra of the original Hopf algebra, when the vector space structure on the quo- tient is altered. We also show the original Hopf algebra is an extension--as an algebra and coalgebra--of the quotient by the primitively generated sub Hopf algebra. Chapter I The study of Hopf algebras is a self-dual theory. For this reason diagram notation is useful, as it makes dual definitions and proofs evident. For all time we fix the field k which is the base for all vector spaces. If X1 ,...,Xn are vector spaces over k , n the permutation group on n-letters a e we consider a: Xe - X -> X ... ex x1@ -0 -@ x0 ->x. (& x, Often a will be written (1 ,... in) where i ,...1 1,2,...n) ; in this case 111n X@-- OX ''' 0y ... e X . 1 n If X X2 1 = n , @Xn is a left G- module. If X, Y are vector spaces "f: X -Y" means f is a linear map from X to Y Once we define a "right" object such as module or co- module, we consider the "left" object to be defined with the mirror definition. Similarly for "left" objects. 8. Algebras (A,m,-q) is an algebra (over k) where A is a vector space over k , m: A @ A -+ A T: k -+ A , if the follow- ing diagrams are commutative. Igm Ti@I A®A A - 3 A @A k(DA~- A®1@A I) m®I m A II) -m m 1S A AA ) A A (1k A AA I) is equivalent to associativity. II) is equivalent to TI(1), (rI) is a unit. k is an algebra where a is the identity, m is the usual multiplication. An algebra A is commutative if AO A m (2,1) A is commutative. A e A If A is an algebra, X, Y vector spaces f: X ->Y then kD@f: X -+ A Y x -+ l@ f(x) where for an algebra A , 1 E A always denotes 'QA(l) If A and B are algebras A & B is an algebra where 9. (A B) q (A @ B) (> A A BOB m A mB mA® B A( B and nA e B = k @ 11B A D k . f: A ->B is a morphism of algebras if the following diagrams are commutative: A A A ff ff k f B B B 4 B -$A k k A will often be written k A If A is an algebra a left A-module is a vector space M with a map ?P: A®D M -.*M satisfying: k@I A (A M )A M M > A (M M A 0 M 'f M If M, N are left A-modules f: M -+ N is a mor- phism of left A-modules if I 10. A M-) M jf is commutative . SI@ f A@N& N > N An augmentation EA of an algebra A is an algebra morphism eA : A -+ k . An augmented algebra is an algebra with a fixed augmentation. If A is ar augmented algebra k is a left (or right) A-module by: EI I m A 0 k - 3 k k -+ k If M is a left A-module and N a right A-module N @A M is a vector space such that *N I - (9 ?M NO A( M ) NO M - N (AM _+ 0 is an exact sequence of vector spaces. If A is an augmented algebra, A+ = Ker e A, M a left A-module, then k ®AM = Coker (A+O M ) A 0 M - ) , i@I ?P or if A+ - M denotes Im(A + M ) A M - M) i®I k A M = M/(A+ - M) Coalgebras (C,d,E) is a coalgebra over k where C is a vector space over k , 11. d: C -+ CtC s: C - k if the following diagrams are commutative: k C ' C C d C C O----C d I) d dO I C II) C®C I d C SC 0C C (k - C@ C I E I) is equivalent to coassociativity II) is equivalent to E is an augmentation of a coalgebra. k is a coalgebra where d = k® I and e = the identity. A coalgebra is cocommutative if C @ C (2,1) C is commutative. C OC d If C is a coalgebra and X, Y are vector spaces f: X - Y , then E f: C®X - Y C D X c k Y ® aYc If C, D are coalgebras C @) D is a coalgebra where 12.
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